%I #83 Mar 08 2024 11:02:03

%S 0,2,6,8,12,16,20,26,30,32,36,42,48,50,56,60,66,70,76,80,84,90,94,100,

%T 110,114,120,126,130,136,140,146,152,156,158,162,168,176,182,186,188,

%U 196,200,210,212,216,226,236,240,246,252,254,264,270,272,278

%N Minimal difference s(n) between beginning and end of n consecutive large primes (n-tuplet) permitted by divisibility considerations.

%C Tony Forbes defines a prime k-tuplet (distinguished from a prime k-tuple) to be a maximally possible dense cluster of primes (a prime constellation) which will necessarily involve consecutive primes whereas a prime k-tuple is a prime cluster which may not necessarily be of maximum possible density (in which case the primes are not necessarily consecutive.)

%C a(n) >> n log log n; in particular, for any eps > 0, there is an N such that a(n) > (e^gamma - eps) n log log n for all n > N. Probably N can be chosen as 1; the actual rate of growth is larger. Can a larger growth rate be established? Perhaps a(n) ~ n log n. - _Charles R Greathouse IV_, Apr 19 2012

%C Conjecture: (i) The sequence a(n)^(1/n) (n=3,4,...) is strictly decreasing (to the limit 1). (ii) We have 0 < a(n)/n - H_n < (gamma + 2)/(log n) for all n > 4, where H_n denotes the harmonic number 1+1/2+1/3+...+1/n, and gamma refers to the Euler constant 0.5772... [The second inequality has been verified for n = 5, 6, ..., 5000.] - _Zhi-Wei Sun_, Jun 28 2013.

%C Conjecture: For any integer n > 2, there is 1 < k < n such that 2*n - a(k)- 1 and 2*n - a(k) + 1 are twin primes. Also, every n = 3, 4, ... can be written as p + a(k)/2 with p a prime and k an integer greater than one. - _Zhi-Wei Sun_, Jun 29-30 2013.

%C The number of configurations that realize this minimal diameter, is A083409(n). - _Jeppe Stig Nielsen_, Jul 26 2018

%D R. K. Guy, "Unsolved Problems in Number Theory", lists a number of relevant papers in Section A8.

%D John Leech, "Groups of primes having maximum density", Math. Tables Aids to Comput., 12 (1958) 144-145.

%H T. D. Noe, <a href="/A008407/b008407.txt">Table of n, a(n) for n = 1..672</a> (from Engelsma's data)

%H Thomas J. Engelsma, <a href="http://www.opertech.com/primes/k-tuples.html">Permissible Patterns</a>

%H Tony Forbes and Norman Luhn, <a href="https://pzktupel.de/ktuplets.php">k-tuplets</a>

%H Daniel A. Goldston, Apoorva Panidapu, and Jordan Schettler, <a href="https://arxiv.org/abs/2208.13931">Explicit Calculations for Sono's Multidimensional Sieve of E2-Numbers</a>, arXiv:2208.13931 [math.NT], 2022. See H(n) in Table 1 p. 2.

%H G. H. Hardy and J. E. Littlewood, <a href="http://dx.doi.org/10.1007/BF02403921">Some problems of 'partitio numerorum'; III: on the expression of a number as a sum of primes</a>, Acta Mathematica, Vol. 44, pp. 1-70, 1923. See final section.

%H Norman Luhn, <a href="https://pzktupel.de/ktpatt_hl.php">Patterns of prime k-tuplets & the Hardy-Littlewood constants</a>.

%H A. V. Sutherland, <a href="http://math.mit.edu/~primegaps">Narrow admissible k-tuples: bounds on H(k)</a>, 2013.

%H T. Tao, <a href="http://michaelnielsen.org/polymath1/index.php?title=Bounded_gaps_between_primes">Bounded gaps between primes</a>, PolyMath Wiki Project, 2013.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PrimeConstellation.html">Prime Constellation.</a>

%F s(k), k >= 2, is smallest s such that there exist B = {b_1, b_2, ..., b_k} with s = b_k - b_1 and such that for all primes p <= k, not all residues modulo p are represented by B.

%Y Equals A020497 - 1.

%Y Cf. A083409.

%K nonn,nice

%O 1,2

%A T. Forbes (anthony.d.forbes(AT)googlemail.com)

%E Correction from Pat Weidhaas (weidhaas(AT)wotan.llnl.gov), Jun 15 1997

%E Edited by _Daniel Forgues_, Aug 13 2009

%E a(1)=0 prepended by _Max Alekseyev_, Aug 14 2015